A characterization of the Wishart exponential families by an invariance property

Abstract

E is the space of real symmetric (d, d) matrices, andS and$\bar S$ are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in

$$\Lambda = \left\{ {\frac{1}{2},1,\frac{3}{2}, \ldots \frac{{d - 1}}{2}} \right\} \cup \left] {\frac{{d - 1}}{2}, + \infty } \right[$$

The Wishart natural exponential family with parameterp is a set of probability distributions on$\bar S$ defined by

$$F_p = \{ \exp [ - \tfrac{1}{2}Tr(\Gamma x)](det\Gamma )^p \mu _p (dx);\Gamma \in S\} $$

where μ_p is a suitable measure on$\bar S$. LetGL(ℝ^d) be the subset ofE of invertible matrices. Fora inGL(ℝ^d), define the automorphismg _a ofE byg _a(x)=^t axa, where^ta is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg _a for alla inGL(ℝ^d) if and only if there existsp in Λ such that eitherF=F _p, orF is the image ofF _p byx↦−x. (Theorem).